Compressive Parameter Estimation for Sparse Translation-Invariant Signals Using Polar Interpolation

Abstract

We propose new compressive parameter estimation algorithms that make use of polar interpolation to improve the estimator precision. Our work extends previous approaches involving polar interpolation for compressive parameter estimation in two aspects: (i) we extend the formulation from real non-negative amplitude parameters to arbitrary complex ones, and (ii) we allow for mismatch between the manifold described by the parameters and its polar approximation. To quantify the improvements afforded by the proposed extensions, we evaluate six algorithms for estimation of parameters in sparse translation-invariant signals, exemplified with the time delay estimation problem. The evaluation is based on three performance metrics: estimator precision, sampling rate and computational complexity. We use compressive sensing with all the algorithms to lower the necessary sampling rate and show that it is still possible to attain good estimation precision and keep the computational complexity low. Our numerical experiments show that the proposed algorithms outperform existing approaches that either leverage polynomial interpolation or are based on a conversion to a frequency-estimation problem followed by a super-resolution algorithm. The algorithms studied here provide various tradeoffs between computational complexity, estimation precision, and necessary sampling rate. The work shows that compressive sensing for the class of sparse translation-invariant signals allows for a decrease in sampling rate and that the use of polar interpolation increases the estimation precision.

Separation of delayed parameterized sources

“…Include sĵ(t − cĵ − η; wĵ) as new column in S The authors in [15], [16] advocate the use of polar interpolation which empirically outperforms other interpolation techniques in terms of delay estimation. This strategy exploits the fact that delayed versions of a source sĵ(t) form a manifold which lies on the surface of a hypersphere (because the 2 -norm of the sources is preserved under delay variations).…”

“…Estimating the source parameters in a unique mixture shares similarities with sparse deconvolution, where time-shifted copies of a perfectly known waveform are gathered into a dictionary [15], [16]. We extend this framework to the case of numerous sources and mixtures, by considering different waveforms (i.e.…”

“…Furthermore, the sparse solution is designed to comply with the delayed source separation model, in particular the fact that sources occur at most once in a mixture. In addition, our approach exploits recent results on sparse deconvolution that allow for continuous delay estimation [15], [16]. Finally, the dictionary does not need to be stored.…”

Compressive line spectrum estimation with clustering and interpolation

“…Therefore, the observation for a frequency value outside of the sampling set Ω can be accurately estimated by its surrounding sampled frequencies using interpolation. Although Taylor series interpolation has been used in this case, certain applications like frequency estimation feature parametric invariance of the norm and distances between signals, and are better suited to a polar interpolation scheme [18,19].…”

A general and yet efficient scheme for sub-Nyquist radar processing

“…With the estimated Doppler shifts, we establish a tactics that further decomposes the estimation of time delays into a series of compressive parameter estimation problems with each corresponding to a distinct Doppler shift. The compressive parameter estimation techniques [3,[9][10][11] can be taken to estimate the time delays. By the sequential and decomposed processing, combined with parametric estimation techniques, the novel estimation scheme has high computational efficiency and high estimation accuracy.…”